Radial finite mass and compactly supported steady states of the asymptotically flat Einstein-Vlasov and Einstein-Euler systems represent isolated self-gravitating stationary galaxies and stars respectively. Upon the specification of the equation of state, such steady states are naturally embedded in 1-parameter families of solutions parametrised by the size of their central redshift. In the first part of the talk we prove that highly relativistic galaxies/stars (the ones with high central redshift) are linearly unstable (joint work with Zhiwu Lin and Gerhard Rein). This is consistent with an instability scenario suggested in 1960s by Zeldovich et al. in the Vlasov case, and Wheeler et al. in the Euler case. In the second part of the talk we explain and prove the Turning Point Principle for the Einstein-Euler system, proposed by Wheeler et al. (joint work with Zhiwu Lin).